However, note that the composition step as is can only guarantee that the expected bias of a long code \(\lbrace f_v\rbrace \)  for a randomly chosen vertex \(v\)  is \(\mu \) , and as such this does not imply the above concentration guarantee that is needed to argue soundness. In particular, in the context of the reduction, this can allow cheating assignments where the adversary can set a subset of long codes to be the constant all ones functions and the remaining to be all zeros, which can cause the above analysis to fail. Therefore, in order to ensure that even under such assignments, the biases of the averaged long codes concentrate around the global bias \(\mu \) , the averaging operator should have good mixing properties, or equivalently, have large spectral gap. This requires the use of non-standard outer verifiers such as the Quasirandom PCP [1]} or SmallSetExpansion [2]}, [3]}. In particular, we shall use the SSE based framework introduced in [3]}.
